11

Theory and Implications of Gravity Probe B

John MesterStanford University

Orbit

22

The Relativity Mission ConceptThe Relativity Mission Concept

!

" = " G

+ " FD

=3GM

2c2R3

R # v ( ) +GI

c2R3

3R

R2$

e% R ( ) &$

e

'

( )

*

+ ,

33

Tests of General Relativity: Background

Einstein Field Equation Gµν = Rµν – 1/2 gµνR = (8πG/C4)Tµν

“matter tells spacetime how to curve, and curved space tells matterhow to move”

No adjustable parameters– G directly measurable Newtonian gravitational constant

Schwarzschild solution: static, spherically symmetric field of a point mass– weak field expansion to first order

ds2 = (1–2GM/C2R)C2dt2 – (1+2GM/C2R)dr2

Φ = GM/R

44

Metric Outside Rotating Earth

The metric outside the rotating earth to lowest non trivial order in Φ and in v -the relevant parameters- involved is given by:

g00 = (1–2Φ/C2)

gij = - δij (1+2Φ/C2)

gi0 = ζi where ζ = 2GIe/C2R3 (R x ωe )

Ie and ωe are the moment of inertia and rotation velocity of the earth

55

Metric Outside Rotating Earth 2Plausibility argument and interpretationWorking to first order (and setting C=1) we start with the line element for astationary point mass ds2 = (1–2GM/R)dt2 – (1+2GM/R)dr2

Generalize to moving point mass by transforming to moving system w/ 1storder Lorentz xform t = t-vx xr = x-vt. (for motion in x direction) ––> ds2 =(1–2GM/R)dt2 – (1+2GM/R)dr2 + 4m/r vdxdtSo for motion in general direction:ds2 = (1–2GM/R)dt2 – (1+2GM/R)dr2 + 4m/r v•dr dt

Since the theory is linear to this order, can superpose distribution of pointmasses to get ds2 = (1–2Φ)dt2 – (1+ 2Φ)dr2 + h•dr dt where,

Thus Φ(r) and h(r) are scalar and vector potentialsin analogy with EM, coulombs law + Special Relativity yields magnetism

See Adler and Silbergleit, IJTP 39 pg 1291, 2000.

!

"(r) =G#dr'

r'$

!

h(r) = 4G"vdr'

r'#

66

Effects of Gravitation on Spin

Particle in Free Fall Uα = dxα/dτ = four velocity dUα

αdτ = 0

Sα = four spin dSα/dτ = 0 in Special Rel

Sα = {S,0} in rest frame Sα Uα = 0 in all frames

To go from Special Relativity to General Relativity make equations covariant––> derivatives go over to:

dUµ/dτ + ΓµνλUνUλ = 0

dSα/dτ – ΓµανUνSµ= 0 Parallel transport

Γσλµ = 1/2gνσ [ dgµν /dxλ + dgλν /dxµ – dgµλ /dxν ] Affine Connection

See Weinberg Gravitation and Cosmology and Misner, Thorn and Wheeler Gravitation.

77

Precession of a Gyro in Earth Orbit

In orbit ––> no forces on gyroTherefore Four spin will undergo Parallel transport

dSα/dτ – ΓµανSµdxν/dτ = 0

Sµdxµ/dτ = 0 ––> S0 = -ViSi

So can eliminate S0 to get:dSi/dt = Γj

i0Sj – Γ0k0VkSi + Γj

ikVkSj – Γ0ikVkVjSj

Define precession vector Ω by dS/dt = Ω x SThen using gµν of earth to calculate Γs and eliminating terms which vanishfor periodic orbits one finds:

Where R is the vector from the earth center to the gyro, v is the gyro velocity

!

" = " G

+ " FD

=3GM

2c2R3

R # v ( ) +GI

c2R3

3R

R2$

e% R ( ) &$

e

'

( )

*

+ ,

88

Precession of a Gyro in Earth Orbit 2

The first term, the geodetic precession, also called the deSitter effect, iscaused by the interaction of the mass or the earth with the gyro frame and thedistortion of spacetime by the mass of the earth and depends on the orbitalvelocity of the gyro

The latter term, the Frame Dragging or Lense-Thirring termdepends on the angular momentum of the earth. The earth is the prime solarsystem body for the source since it’s angular momentum is well known.

Since its an interaction between spin angular momenta of the earth and gyrothis term is sometimes referred to as ‘hyperfine interaction’, in analogy withatomic physics. But does not depend on magnitude of gyro spin.Any vector that points a fixed direction in the local inertial frame will precessin frame of distant stars.

!

" G

=3GM

2c2R3

R # v ( )

!

" FD

=GI

c2R3

3R

R2#

e$ R ( ) %#

e

&

' (

)

* +

99

Precession of a Gyro in Earth Orbit 3For a polar orbit the Frame dragging and geodetic precession areperpendicular and can therefore be resolved independently

Both terms are maximized by taking Rorbit close to Rearth

For 642km altitude and gyro spin near the equatorial planeThe magnitude of the effects averaged over an orbit are

Geodetic ΩG = 6.6 arcsec/year Frame Dragging ΩFD = 0.041 arcsec/yearGP-B goal is to test these to parts in 105 and to better then 1% resp.

Leading Corrections are due to:Geodetic effect due to the sun, ΩG

s = 19 marcsec/year(in plane of the ecliptic)

Oblateness correction of Geodetic Effect, ΩGJ2 = –7marcsec/year

(parallel to ΩG)

Frame dragging due to the sun and effects from the moon are negligible

1010

Interpretation of Predicted EffectsAs noted in the introduction, General Relativity is a theory with no adjustableparameters. So there is no need for interpretation - all of the science of GP-B isrevealed by the precession equation

Still it can be interesting to form a physical picture of how the precessions arise.But these pictures are probably a matter of taste. So if a particular interpretationdoesn’t appeal to you, just go back to the equations.

!

" = " G

+ " FD

=3GM

2c2R3

R # v ( ) +GI

c2R3

3R

R2$

e% R ( ) &$

e

'

( )

*

+ ,

1111

Parameterized Post-Newtonian Formalism PPNEddington, Nordtvedt, Will

• Useful parameterization for a range ofmetric theories of gravity

• Low velocity, weak field limit– Solar system

• Metrics of various theories similarform– Expansion about flat (Minkowski)

space– Gravitational potentials provide

deviation from flat space– Various theories provide different

coefficients

0Conservationof momentum

Γ1−4

0Preferredframe

α1−3

0Preferredlocation

ξ

1Non-linearity ingravitysuperposition

β

1Spacecurvature

γ

GR valueMeaningParameter

!

" FD

= # +1+$1

4

%

& '

(

) *

GI

2c2R3

3R

R2+

e, R ( ) -+

e

.

/ 0

1

2 3

!

" G

= # +1

2

$

% &

'

( )

GM

c2R3

R * v ( )

Caveat: Danger in viewing all GR tests in terms of PPN parameter limits. Metrictheories that fit within PPN framework span only a part of the space of possiblealternative theories. Most recent attempts at unification of the forces of naturerequire non-metric revisions of GR.

1212

Significance of Frame DraggingIn GR gravitational effects are due to the geometry of spacetime.A non-rotating body produces a static spacetime curvature due to Mass. Arotating body produces an additional dragging of external inertial frames viathe body’s angular momentum, analogous to the body being surrounded by aviscous fluid. The instantaneous direction change of the gyro over the polesin opposite in direction to that when the gyro is above the equator. This isjust what a test “straw” oriented perpendicular to the immersed in a fluidwould do.Frame Dragging is conceptually related to Mach’s principle: Inertial forcesarise from accelerations and rotations with resp. to total mass of the universe.Under this interpretation the gyro spin direction reaches a compromisebetween following the distant stars and the rotating earth.Lense and Thirring in 1918 considered the metric inside a rigidly rotatinghollow sphere. In 1966 it was shown that as the shell’s gravitational radius,2GM/c2, approaches its physical radius, the precession of a gyro at the centerapproaches the angular velocity of the shell. That is, the gyro spin becomeslocked to the rotating distant matter. Water would climb up the sides of anon-rotating bucket at the center of the shell.

1313

Gravitoelectric / Gravitomagnetic ViewpointSpace-time

metric

Newtonian

analog

EM analog Gravito

EM analog

Rotational

effect

g00 ! V Eg 1/3"G

g0i No analog Ai Bg "FD

gij No analog No analog No analog(space curvature)

2/3"G

This view point arises in analogy with electromagnetism or fromtreating GR as a spin 2 field theory to lowest order. Static mattergenerates a gravito-electric potential and space curvature. Rotationproduces a gravito-magnetic dipole field. Frame dragging arisesform the interaction of the gyro spin with a vector potential g0i.

A moving mass (mass current)generates a gravito-magnetic field; itis related to gravito-electrostatics through Lorentz xform (see slide 11).However, a gravito-magnetic field of a rotating mass cannot bederived by rotation of coordinates from the static field of a non-rotating mass. Frame Dragging as tested by GP-B does not have atrivial relationship to static gravitation.

1414

Evidence of Frame Dragging EffectsWithin GR, frame dragging should produce a precession of the orbital plane ofa body in earth orbit (the original prediction of Lense and Thirring) and aprecession of the pericenter of the orbit. Cuifolini et al. have analyzed laserranging data of the LAGEOS I and LAGEOS II satellites and claim aconfirmation of FD effects to a 25% level. Results are controversial due to theneed to subtract out Newtonian perturbations that are > 107 times larger thanthe GR predictions, as well as understanding non-gravitational disturbancesdue to radiation pressure and thermal effects.

In the Taylor-Hulse binary pulsar B1913+16, uncertainties in the constituentproperties are too large to be used for precision tests of gravitomagneticeffects. A pulsar orbiting a fast rotating black hole could provide an appropriatetest system; no definitive candidates are yet know.

Some effects of frame dragging by translational motion are present (incombination with other effects) in Lunar Laser ranging tests of the Nordtvedteffect.

Within the PPN framework, Shapiro time delay and LLRmeasurements place tight limits on the γ and α1 parameters.

1515

Frame-Dragging Evidence in Astrophysics

Observed jets from Galaxy NGC 6251 Power output 1038 watts!(A trillion suns)

Compact object holds jet directionaligned millions of years. Plausiblecause of the alignment is the effect offrame dragging from a SupermassiveBlack Hole. “Frame dragging imprintsthe angular momentum of the source onthe distant spacetime.” Evidence for FDbut does not allow for quantitativemeasurement.

Radio Map ofGalaxy NGC 6251

1616

Interpretation of Geodetic Precession

The Geodetic effect is a combination of precession due the gravito-electricfield and precession due curved spacetime.

The Missing Inch(and a half)The gravito-electric part, 1/3 of the total effect, is due to special relativisticcorrections to the motion of the gyro in the g00 potential.Consider the path traveled by the gyro as it goes around the earth at velocityv. From special relativity the distance traveled is contracted:

So for assumed circumference of the orbit,L0 = 2π(Re+Alt) = 2π x7010km = 44,228kmAnd gyro velocity v~L0/97.6 min = 7553 m/sec

So L0 – L = 0.014m (about half an inch)

!

L = L01"

v2

C2

1717

Interpretation of Geodetic Precession 2

The other 2/3 of the precession is due the curvature of space around theearth. Change in vector orientation under parallel transport is a hallmark of curvature as seen in 2 dimensional example.Transporting from points Q-P-N-Q the vector is rotated.

What about the missing Inch?For the GP-B case the curvature is due to the mass ofthe earth distorting the space around it. This producesan orbit circumference smaller than what one wouldcalculate using 2πr in flat space, smaller by about 1 inch over 27000 miles.

A B

B’

The effect this has on gyro spin direction can bevisualized with the diagram due to Kip Thorn. As thegyro orbits about point A the spin direction staysfixed in the local flat sheet geometry. Cutting out thegap wedge and connecting points B and B’ to createa cone one sees a change in spin direction.

For GP-B the 1.5 inch gap corresponds to ~1marcsper orbit ==> 6.6 arcsec over a year.

}1.5” gap

1818

Evidence of Geodetic Effects

For The Earth-Moon system can be considered as a ``gyroscope'', with itsaxis perpendicular to the orbital plane. The predicted precession, firstcalculated by de Sitter, is about 19 marcseconds per year. This effect hasbeen measured to about 0.7 percent using Lunar laser ranging data.

Within the PPN framework, light deflection and Shapiro time delaymeasurements around the sun set limits on the curvature parameter γ.The current limits established by telemetry data from the Cassini spacecraftare a few parts in 105.

Several alternative theories, such as Damour-Nordtvedt "attractormechanism" tensor-scalar theories of gravity, predict deviations from zeroof the PPN parameter γ of up to γ-1≤ 4×10-5 )

1919

References

Will, Confrontation between General Relativity and Experiment, Living Reviews inRelativity, lrr-2001-4 http://relativity.livingreviews.org

Adler and Silbergleit, General Treatment of Orbiting Gyroscope Precession, IJTP 39 pg1291, 2000

The National research Council Space Science Board, Review of Gravity Probe B,National Academy Press, 1995

Weinberg, Gravitation and Cosmology. J Wiley and Sons

Misner, Thorn and Wheeler, Gravitation.